Why do ions flow from high concentration to low concentration?

This a question related to ATP pump that uses electrochemical gradient to provide energy to make ATP. I have few questions about this.

1. In an electrochemical gradient there are more positively charged ions on one side of the membrane(In proton gradient case). So why do the protons move from highly concentrated area to low concentrated area? Is it because the side that has more protons repel each other and eventually push the protons away until the charge is neutralized?
2. If there is a higher concentration of ions in one area why do they move to the less concentrated area? Is it because higher concentrated area has higher pressure so they move to lower pressure area, because that side provides less impedance.
3. Now what exactly gives the energy for the ATP pump. Is it the movement of ions. Is it using the kinetic energy of the movement of ions across a gradient to produce ATP? If so why does it need a proton gradient, ions are moving all the time randomly even without gradient? Does the gradient give it a direction or something that can be efficiently utilized the pump.

I'd even compare it to a gas where there is no interaction at all! If you have a higher concentration on one side, there will be more molecules crossing the barrier from this direction than in the opposite direction. Just because there are more candidates attempting to cross the side. As soon as the concentrations are equal, this effect stops.
Note the whole time there was absolutely no interaction between any of the molecules. So a single particles doesn't even know about any of the others.

where dC is the concentration gradient, Z the electric charge of the ion, and E the electric field. At equilibrium, dG = 0 and so there is a balance of concentration graident and electric field (hence the membrane potential):

For many ions (protons across the mitochondrial membrane, Na+, Ca++, K+, etc) the concentrations follow the setpoint of the membrane potential (60 mV for the cell membrane, 220 mV for the mitochondrial membrane). That's one way to think of the membrane potential- it stores energy by segregating ions. The change in free energy as an ion crosses the membrane (via ATP synthase, for example), is used by the cell to do something useful.